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% 标题
\title{Fight with Wildfires}
\begin{document}
% 摘要
\begin{abstract}
	
	How to purchase drones to meet the requirements of mountain fire emergency rescue? In our analysis, we examine the minimum cost of drones for surveillance and situational awareness (SSAD) and radio repeater drones (RRD), for the demands of government’s purchase of emergency drones.

	First, we present a model covering the whole Victoria state, dividing Victoria state into 23 * 31 squares. Thus we can determine the radio propagation distance and the number of fire points within the command range of each EOC. Through research, we find that the terrain has a great influence on the number of RRDs. The more rugged the terrain, the shorter the distance the radio signals travel, and the more RRDs are needed to amplify the radio signals. When the scale of the fire is larger, in order to ensure the personal safety of frontline personnel, there will be more fire spots, then the research in this paper shows that the more drones are needed. The total number of SSADs is 7548 and the total number of RRDs is 2706, at a cost of 10,254,000 AUD.

	Next, we set the mixed integer nonlinear programming model for SSAD and RRD respectively. After processing the data of Victoria, we obtain the location and number of watchpoints of each square. Then we are able to investigate the number and trajectory distribution of both SSAD and RRD in each square grid by our model. The replacement of drones for continuous working is also taken into consideration. Through research, we find that the flight time of SSAD is shorter than the charging time, so each flight line needs more spare drones. The number of RRD is greatly affected by the distribution of fire points.

	Finally, we set up a fitting model for the future forecast, fitting the data of the past 20 years. Through this process, a group of certain functions is obtained, which can be used for Victoria’s s fire prediction in the next 10 years. From the forecast results, our model and the demand for the next decade adapt very well. Then those prediction data are analyzed together with the previous dataset to obtain a possible distribution of fires to those 23*31 squares. Then use this new distribution to feed our models of calculating drones’ numbers, thus we can obtain a final result. Besides, considering the loss of battery, Ten years later, the cost will increase by 308,000 Australian dollars.

	Our model has good adaptability and helps reduce expenses. By tuning the parameters, our model can adapt to other landscapes and the sudden extreme fires. And in our model, different situations have been taken into consideration to minimize the number of each kind of drone. In the meanwhile, we manage to guarantee enough drones for a fire emergency.

\end{abstract}

% 目录页 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle         % 控制序列
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\newpage

\section{Introduction}
Wildfires caused by nature occur all over the world every year, some of which are destructive and unavoidable. Australia has suffered a serious wildfire attack in 2019-2020, which has caused serious damage to the life safety of local residents and the natural environment. In order to minimize the loss of life and property caused by wildfire, many new fire-fighting equipment have been put into use. On the one hand, the combination of wearable devices and SSA UAV makes it possible for EOC to monitor the life condition and fire situation of front-line firefighters in time. On the other hand, the wide use of portable radio equipment also makes it easier for the front line to communicate with EOC.

However, there are still some factors that make it difficult to apply these methods in practice. For example, the high price of special UAVs, short battery life and long charging time make it difficult for UAVs to put into use in large quantities. In addition, the low power of portable radio equipment also makes the communication distance and stability have a big test. The terrain also has a great influence on the radio propagation distance.

\subsection{Problem Summary}
\begin{itemize}
	\item The range of handheld radios is limitted by their low transmitting power, varying from \SI{2}{km} over flat area to \SI{5}{km} over rugged mountainous area.
	\item Repeaters carried by hovering drones can extend radio range to \SI{20}{km} and hardly be influenced by topography or other factors.
	\item SSA drones can monitor and report data from firefighters' wearable devices, which is of great significance for situation evaluation and safety guarantee.
	\item Each UAV can travel up to \SI{30}{km}, at a max speed of \SI[per-mode=symbol]{20}{\metre\per\second} within \SI{2.5}{\hour}, no matter what kind of equipment it carries.
	\item 1.75 hour recharge time for built-in UAV battery is required after each flight.
	\item Each UAV equipped with a redio repeater or an SSA facility costs \SI{10000}{AUD}.
	\item Auxiliary batteries for radios or SSA can be swapped while the built-in battery recharges.
\end{itemize}


\section{Assumptions}
\subsection{Model Assumptions}
\begin{itemize}
	\item SSAD uses its own cameras and telemetry sensors to monitor and report the data of frontline personnel's wearable devices, then store them in its own storage devices, finally report the monitoring data after returning to EOC.\@
	\item SSAD will not exchange data information with each other or RRD.\@
	\item Regardless of the time spent by the SSAD in monitoring the frontline personnel, the SSAD immediately completed the data collection of the wearable device of the frontline personnel around a fire point when it passed this fire point.
	\item All SSADs fly in the same horizontal plane.
	\item The trajectory of each SSAD is fixed.
	\item There are no overlapping fire points between each two different trajectories, and the only overlapping point between them is EOC.
	\item SSAD flies at a constant speed throughout the flight.
	\item Sfter an SSAD returns to EOC, the next SSAD along the same trajectory will depart immediately.
\end{itemize}


\subsection{Nomenclature}

\begin{table}[H]
	\caption{Nomenclature}
	\label{tab:3}
	\centering
	\begin{tabular}{ll}
		\toprule
		symbols & definitions                       \\
		\midrule
		SSA & surveillance and situational awareness \\
		SSAD & SSA drone  \\
		RRD & radio repeater drone   \\
		\(S\)   & fire point, or source         \\
		\(D\)   & destination of drone        \\
		\(A\)   & SSAD, or flight track of SSAD        \\
		\(N_S\)   & the number of fire points $S$ \\
		$\mathcal{S}$ & a set consists of all S \\
		$\mathcal{A}$ & a set consists of all A \\
		$\alpha$ & the number of fire point on a flight track \\
		$T$ & the flight time of each circle \\
		$T_{\max}$ & the maximum flight time \\
		$N_a$ & the number of flight track \\
		$v$ & velocity \\
		$v_{\max}$ & the maximum velocity \\
		$num_{sb}$ & the maximum number of observation points in block \\
		$num_{se}$ & the number of areas sharing the same equipment \\
		$f_T$ & the fire frequency threshold to determine the need of sharing \\
		\bottomrule
	\end{tabular}
\end{table}

\section{Problem 1}

We can see from the question that WileE–15.2X Hybrid Drone can only carry either radios or video/telemetry. The two need to work in completely different ways and cannot be replaced by each other. So SSA drones and Radio Repeater drones are completely independent, referred to as SSAD and RRD respectively. SSAD is responsible for monitoring and reporting the situation of frontline personnel, and RRD is responsible for realizing frontline personnel's handheld two-way radio and EOC signal relay. The built-in battery of the drone is not removable and needs to be returned to the EOC to charge it. However, the batteries of cameras and radios can be removed. Assuming that the battery life is greater than 2.5 hours, it can be guaranteed that the radios, videos and telemetries are always powered. When calculating how many drones need to be equipped at the beginning, the loss of the drone's built-in battery is not considered for the time being.

We assume that each aircraft is regarded as a point and will not collide with each other. Assuming that each firefighter is distributed close to a certain fire point, and each fire point has a team leader responsible for communication. When the UAV is located at each fire point, it can receive all the information of the frontline personnel around the fire point.

\subsection{Map Division}

Because the UAV’s battery needs to be returned to the EOC for charging, the command range of each EOC is half the maximum flight distance of the drone, which is 15km.

Our model first selects a suitable rectangle containing Victoria, where the latitude and longitude coordinates of a pair of vertices are (140°53'E, 39°31'S), (150°E, 34°S). Divide the rectangle into 23*31 squares, where the area of each square is approximately the command range of each EOC. In each area, only one EOC is needed. This EOC can direct the drone to any point in the area for monitoring and relay. In the process of data processing, we only make the data in the Victoria boundary line valid, and the data in the remaining rectangles are uniformly set to zero.

\begin{figure}[H]
	\centering
	\includegraphics[scale=0.5]{grided vic.png}
	\caption{Gridded Victoria Map}
	\label{fig:1}
\end{figure}

\subsection{Each Square Area}

\subsubsection{Assumptions of SSAD for Each Square}

The responsibility of SSAD is to fly to the front line, monitor the data of front-line personnel wearable devices and fly back to report to EOC, thereby helping EOC know the front-line situation and command the front-line personnel in time.

Our model assumes that SSAD obeys the following rules:
\begin{itemize}
	\item SSAD uses its own cameras and telemetry sensors to monitor and report the data of frontline personnel's wearable devices, then store them in its own storage devices, finally report the monitoring data after returning to EOC. SSAD will not exchange data information with each other or RRD. Regardless of the time spent by the SSAD in monitoring the frontline personnel, the SSAD immediately completed the data collection of the wearable device of the frontline personnel around a fire point when it passed this fire point. All SSADs fly in the same horizontal plane.
	\item The fire point is the information source, denoted by S. EOC is the destination of the drone carrying information, denoted by D. SSAD is denoted by A.
\end{itemize}

\begin{figure}[H]
	\centering
	\includegraphics[scale=1]{传图.pdf}
	\caption{S-A-D Relationship}
	\label{fig:2}
\end{figure}

\begin{itemize}
	\item According to the working characteristics of SSAD, we use the Mixed-Integer NonLinear Programming (MINLP) model to obtain the optimal allocation plan of SSA drones.\cite{wu2}\cite{wu2020research}
\end{itemize}

\subsubsection{Flight Track of SSAD}

Assuming that a fire has occurred in this square area, $N_S$ represents the number of fire points $S$ within the jurisdiction of an EOC, and the set $\mathcal{S}$ consists of the coordinates of each fire point:
\[
	\mathcal{S}=\left\{S_{1}, S_{2}, ..., S_{N_S}\right\}, 
\] 
SSAD is represented by $A$, $N_{a}$ is the number of SSAD flight tracks, and the set $\mathcal{A}$ is composed of Na flight tracks:
\[
	\mathcal{A}=\left\{A_{1}, A_{2}, \dots, A_{N_a}\right\},
\]
The flight track of each operation of SSAD is represented by the coordinates of the fire point or EOC that SSAD passes through in turn. The flight track of each SSAD is expressed as
\[
	A_{n}=\left(S_{a\left(n, 0\right) }, S_{a\left(n, 1\right) }, S_{a\left(n, 2\right) }, \dots,S_{a\left(n, \alpha_{n}\right) }, S_{a\left(n, \alpha_{n}+1\right) }\right), where\ n=1,2,\dots,N_{a},
\]
Elements such as $S_{a\left(n, \alpha_{n}\right) }$ correspond to the coordinates of a point in $\mathcal{S}$ respectively.

SSAD must start from EOC and finally return to EOC, so
\[
	S_{a\left(1,0\right)}=S_{a\left(2,0\right)}=\ldots=S_{a\left(N_{a},0\right)}=D,
\]
\[
	S_{a\left(1,\alpha_1+1\right)}=S_{a\left(2,\alpha_2+1\right)}=\ldots=S_{a\left(N_{a},\alpha_{N_{a}}+1\right)}=D.
\]

Assume that the trajectory of each SSAD is fixed. There are no overlapping fire points between different trajectories, and the only overlapping point between different trajectories is EOC:
\[
	A_i\cap\ A_j=\left\{D\right\},\ \forall\ i\neq j\in\left\{1,\ 2,\ \ldots,\ N_a\right\},
\]
\[
	\bigcup_{k=1}^{N_a}A_k=\left\{D\right\}+\mathcal{S} ,
\]

\subsubsection{Time Delay of Fire Points' Data}

Assume that the SSAD flies at a constant speed throughout the flight. The SSAD flight time is less than or equal to $T_{max}$, and the flying distance is less than or equal to $d_{max}$, so it is assumed that the SSAD is flying at the speed of $v_{max}$ throughout the entire flight. In addition, the flight time of each circle of SSAD is limited by the Maximum flight time, and the flight distance is limited by the Flight range:
\[
	0<T_n\le\ T_{max},\ \forall\ n=1,2,\ldots,N_a,
\]
\[
	0<v_nT_n\le\ d_{max},\ \forall\ n=1,2,\ldots,N_a,
\]
\[
	N_a\in N^+.
\]

When the SSAD flies to a certain fire point, the data collected here. When the SSAD is getting farther and farther away from this point, the collected information becomes more and more lagging, and there is a time delay in the fire point information received by the EOC. The sum of the delay of all fire points on each track is expressed as
\[
	t_{A_ndelay}=\sum_{k=1}^{\alpha_n}{\frac{|S_{a\left(n,k+1\right)}-S_{a\left(n,k\right)}|}{v_{max}}k},\ n=1,2,\ldots,N_a,
\]
where $|S_{a\left(n,k+1\right)}-S_{a\left(n,k\right)}|$ represents the Euclidean distance between the $\left(k+1\right)th$ fire point and the kth fire point in a certain trajectory.

In the emergency command of forest fires, the timeliness of frontline information is very important. Therefore, we need to control the average value of the delay of each fire point data from collection to transmission to EOC within a fixed range:
\[
	\bar{t}_{delay}\le t_{\max{delay}},
\]
\[
	\bar{t}_{delay}=\frac{\sum_{k=1}^{Na}t_{A_kdelay}}{N_{s}}.
\]

\subsubsection{Sampling Period of Fire Points' Data}

Suppose that after an SSAD returns to EOC, the next SSAD along the same trajectory will depart, that is, only one SSAD is flying on each trajectory in the area outside the EOC at any time. Therefore, the sampling period of the fire point data on a certain trajectory is equal to the time it takes for the SSAD on this trajectory to depart from EOC to return to EOC. Sampling period of each fire point information
\[
	T_n=\frac{\sum_{k=0}^{n}\left|S_{a\left(n,k+1\right)}-S_{a\left(n,k\right)}\right|}{v_{\max}},\ n=1,2,\ldots,N_a.
\]

In the emergency command of forest fires, frontline information needs to be updated frequently. Therefore, the data of each point needs to be sampled frequently. We control the average value of the sampling period of the data of each fire point within a fixed range:
\[
	\bar{T}\le{\bar{T}}_{\max},
\]
\[
	\bar{T}=\frac{\sum_{k=1}^{N_a}{T_k\alpha_k}}{N_s}.
\]

\subsubsection{SSAD Cost of Each Square}

For the UAV equipment demand model when a fire occurs in each area, we use UAV cost as an indicator to measure whether the allocation of SSAD is reasonable. When an SSAD on the kth track flies back to the EOC for charging, in order to ensure that another SSAD takes off immediately, $\left\lceil \frac{t_{charge,\ k}}{T}\right\rceil $ spare SSADs are required, so a total of $\left\lceil \frac{t_{charge,\ k}}{T}\right\rceil+1$ SSADs is required for each track. To ensure the lowest cost, the sum of all SSAD numbers of $N_a$ orbitals should be the smallest:
\[
	\min{\sum_{k=1}^{N_a}\left(\left\lceil\frac{t_{charge,k}}{T}\right\rceil+1\right)}.
\]

\subsubsection{Result of SSAD}

\begin{figure}[h]
	\centering
	\includegraphics[scale=1]{ff3.pdf}
	\caption{Result of SSAD}
	\label{fig:6}
\end{figure}

As we can see in Figure~\ref{fig:6}, the lines with different colors are different routes with the optimal costs. We find that it is clean and pretty, and the routes are reasonable.

\subsection{RRD for Each Square}

Because the coverage of RRD is larger than the longest distance it can reach, we only need to consider whether RRD is within the communication range of fire point. In addition, due to the limited battery life of RRD, it can only stay at the destination for a specific time before returning. Assuming that the destination needs RRD for continuous signal transfer, arrange a schedule for multiple RRDs. In order to save money, we need to solve the minimum number of aircraft.

As is known that the maximum flight time is 2.5 hours, We want RRDs to spend as little time on their journey as possible. Thus, they always travel at the maximun speed of 20m/s. In order to ensure that when one RRD has no power and has to return, the communication will not be interrupted. At this time, it is necessary to ensure that another RRD arrives to take over. 

\begin{figure}[H]
	\centering
	\includegraphics[scale=1]{充电时间的计算.pdf}
	\caption{Distance - time diagram of RRDs}
	\label{fig:3}
\end{figure}

As shown in Figure~\ref{fig:2}, the yellow line and the green line represent two RRDs respectively. When RRD 1 starts to return, RRD 2 must have reached the target position. If RRD 1 starts recharging immediately after returning, and if the recharging time
\[
	T_{recharge}=T_{\max}-\frac{4r}{v_{\max}}
\]
is longer than the actually needed time
\[
	T_{needed}=\SI{2.50}{\hour},
\]
RRD 1 will be able to take over RRD 2 and reach the target position when RRD 2 starts returning. In this case, only two RRDs are needed for the target location to meet the requirement of continuous rotation. Otherwise, three RRDs are required. That is when the distance between EOC and target position $r\le\SI{13.5}{km}$, two RRDs are required. If not, three RRDs are required.

Based on the previous analysis, we make the following assumptions:
\begin{itemize}
	\item The hovering position of RRD does not change.
	\item Each RRD hovering position is operated by several fixed RRDs in turn.
	\item RRDs always fly in a straight line at maximum speed during rotation.
	\item RRDs are very small, so they don't collide in the air.
	\item For a certain block area, the coverage distance of portable radio is the same.
\end{itemize}

\subsubsection{Target Point of RRD}

\begin{figure}[H]
	\centering
	\includegraphics[scale=0.55]{ff1.pdf}
	\caption{Fire points and their maximum communication range}
	\label{fig:4}
\end{figure}

As can be seen from Figure~\ref{fig:4}, each colored circle represents the communication range of a fire point. The point on which the circles intersect can connect to two fire points at the same time. So the relay point is selected as the point closest to (0, 0) in the intersecting area. If a circle does not intersect with other circles, the repeater locates at the point closest to (0, 0) on the circle.

\subsubsection{Result of RRD}

\begin{figure}[H]
	\centering
	\includegraphics[scale=0.85]{ff2.pdf}
	\caption{The Optimal repeaters' location (Marked by Deltas)}
	\label{fig:5}
\end{figure}

As we can see in Figure~\ref{fig:5}, the Delta markers are the optimal repeters' location. 

\subsection{The Number of Drones That CFA Needs to Purchase}

\section{Problem 2}

\subsection{Fire data preprocessing}
We applied for the fire data set of Australia from 2000 to present from the NASA website. The longitude, latitude, time, and fire radiation power information in the Victoria area in the data set are useful for our model. Corresponding effective data information to each map block, after processing, we get 23*31 sets of data information. We use the month as the basic unit, count the number of fires observed in a month as the fire frequency, and average the fire radiation power according to the number of times as the intensity. Each group of information contains information on the frequency and intensity of fires from 2001 to 2020.


\subsection{Fire forecast in Victoria}
We observed the situation of various map blocks in Victoria and found that in the 240 months after processing, some places had fires only a few months, and some places had serious fires every few months. After observing these two common fire frequency distributions, we fit and predict the data for each map block by year. After testing various methods such as gray prediction model, polynomial fitting, Fourier fitting, etc., it is found that the fitted curve does not fit the original data or has a growing trend that does not conform to the situation. So we use a slower-growing custom function, $f\left(x\right)=a\cdot sigmoid\left(x\right)+b\cdot x+c$ to fit and predict the next 10 years.

Selecting the fitting situation of a certain map block, it can be seen from the above figure that the frequency and intensity of fires that have occurred fluctuates about a certain value. The same conclusion was reached after observing multiple map blocks, so we finally chose a custom function prediction value.


\subsection{Setting of fire points in Victoria}

The predicted fire situation in 2021 is selected as the data of this part of the model, the number of observation points is set according to the fire intensity, and we assume that the maximum number of observation points in the map block is 8, that is, each map block contains 0 to 8 points. The number of observation points can be used by models in section3  to obtain the number of SSAD and RRD. In order to save money, we share SSAD and RRD in areas where the frequency of fires is less than a certain number. Assuming that every five areas share a set of equipment, the number of equipment is determined by the largest number of observation points in these areas. Therefore, in data processing, we can update the equivalent distribution of the number of fire points. Then determine the number of drones through section3.

In the case of extreme weather, you can increase the number of observation points in the model, or reduce the fire frequency threshold in the area where the equipment needs to be shared. Thereby, the number of equipment in different regions can be increased in a basis to deal with sudden extreme fires, so that expenses can be saved as much as possible.

In this section, data from September 2018 to September 2019 is selected. Because we assume Victoria suffers from the most serious fire disaster during this period by certain criteria in our model.

In our model, about three parameters are likely to affect the total number of observation points. Those are the maximum number of observation points in each area $num_{sb}$, the number of blocks that share the same equipment $num_{se}$, and the fire frequency threshold $f_T$ for determining the sharing block groups. We obtain the following three figures by using the control variable method, respective influence to final results is analyzed in this section.

\begin{figure}[H]
	\centering
	\includegraphics[scale=0.6]{f123.pdf}
	\caption{Three Figures}
	\label{fig:3}
\end{figure}

With parameters changing within a limited range, we find that only $num_{sb}$ has a significant influence on the final result. When it varies from 6 to 10, the total number of observation points increases significantly. The curves’ shape in picture appear to be similar, and when focusing on a one-time point, the relative increases are quite close. This is because the relationship between $num_{sb}$ and the total number of fire observation points in our model is approximately a linear type.

In the second picture, the frequency threshold $f_T$ is set to $n$ times the maximum fire frequency. We discover that within the tuning range, $f_T$ has little effect on the final result. We speculate that the number of areas that are arranged to share devices grows with $f_T$. Since we choose the largest quantities as a reference, the number of total shared devices would balance the reduction of other distinctive groups’ numbers. Then the total number of observation points remains relatively unchanged.

When the number of areas sharing one set of equipment increases from 3 to 5, the curves in the third picture have almost no difference in years with fewer total watchpoints. However, when the total number rises to a relatively high point, the fewer neighbor blocks sharing one set of devices, the small result tends to be.

The above results indicate that $num_{ab}$ and $num_{sb}$ will affect the final number of observation points, thus affecting the number of dranes.

By adding the maximum number of single domain’s observation points, or reducing the number of areas sharing one set of equipment, the storage quantity of devices can be updated to a bigger one. Under the influence of humidity, temperature, and other factors, the possibility of fire is changing. To deal with the sudden extreme fire disaster, we can artificially tune the above parameters, then purchase new drones based on the results our model obtains. At the same time, we also manage to cut the budget as related algorithms are used in our model.

\section{Problem 3}
\subsection{Change of effective communication distance}
\subsubsection{Problem Analysis}
In equipment for responding to fire emergencies, RRD and handheld radios transmit information in the form of waves. However, in obstacles or complex terrain, the transmission distance of waves will be affected, that is, the effective communication distance will be reduced.

If the communication distance of the RRD is affected by these factors, you can increase the RRD for transfer. Use our Nesting Radio Repeater Drones Model to get the least number of RRDs, and then use our repeater replacement model to calculate the number of replacements.

\subsubsection{Conversion between terrain and communication distance}

In order to get an effective communication distance, we measured the roughness of the terrain. Assuming that the communication only exists inside the map block, through the downloaded terrain information of Victoria, we obtain the variance of the terrain height of each map block to be equivalent to the degree of ruggedness. Finally, normalization is performed to obtain a coefficient matrix of $\beta_{(23,31)}$ to measure the degree of ruggedness.

\subsubsection{Result of Problem 3}

Actually, the result has been mentioned as Figure~\ref{fig:6} and Figure~\ref{fig:5} because our solution to problem 1 is based on solution 3.

\section{The Model Results}

The figure above shows the result of the analysis of Victoria's fire and landscape datasets. We discover that the curves of RRD and ASSD's numbers are quite similar except for a relatively stable gap in each year, although we have constructed separate models for RRD and ASSD. Those three curves are all floating up and down at an almost fixed value. We approach the fixed value by acquiring the mean number of drones. Then for the past 19 years, those average values are 2623, 7137, 9760 for RRD, ASSD, and total drones respectively.

When in the process of prediction, we create a function that grows quite slowly but stabilizes at a certain place, which appears to be around the mean value. After taking this smooth predicting data into our model, we obtain a quite stable result. It is a fixed value for both RRD and ASSD, corresponding numbers are 2706, 7548.

\section{Conclusions}

After dividing Victoria into small pieces, obtaining algorithms of minimizing the number of SSAD and RRD, fitting and predicting fire trend, we can integrate our models into one system. The Victoria datasets are utilized for base settings, then our algorithms for approaching the minimum number of planes lead to an approximate solution for each year. The fitting and predicting part is regarded as constraints to form new datasets that directly determine the trend of our final result.

This system can return the results if it is feed with specific data. The input data are Victoria's data of landscape and fires, and the output is the number of SSAD, RRD, and the total number of drones every year, from September 2001 to September 2030.

Our model can adapt to analyze different landscapes, adjusting the number of drones according to the potential fire by tuning the related parameters. And the algorithms used in the model enable the government to cut expenses.

\section{Strengths and weaknesses}

\subsection{Strengths}
\begin{itemize}
	\item \textbf{Comprehensive}\\
	The model can effectively solve the UAV cost problem considering factors such as flight capability, monitoring needs. The establishment and solution of our model are reasonable and reliable.
	\item \textbf{Flexible Methods}\\
	In the first problem, Mixed-Integer NonLinear Programming (MINLP) is used to quickly obtain the optimal solution. In the first problem, the two objective functions with smaller weights in the multi-objective are converted into a constraint form. So it is more convenient to get the optimal cost. In the second question, we use fitting to predict. Fitting can better predict system restricted by uncertain factors, such as the frequency and magnitude of fire occurrence in this article.
	\item \textbf{Solve Problems in Blocks}\\
	The model provides a thorough explanation of how the cost of drones is affected by various factors. In addition, the analysis of the entire Victorian state will be decomposed into many EOC command blocks. The blocks will be analyzed separately, which expands the scope of application of the model.
	\item \textbf{Big Data Analysis}\\
	Our model uses detailed statistical analysis of the real fire data in Australia observed by NASA. The real data makes the parameter setting more reasonable and the decision-making process more scientific.
	\item \textbf{Think Differently}\\
	When the model solves the problem that requires the number of RRDs, it is not based on the traditional idea of selecting relay points first and then gradually reducing. It first draw circles according to the communication distance of each fire point, and find the intersection between the circles to determine the location of repeaters. The model is planned with the use of graphic nature, which is exquisite.
	\item \textbf{Useful}\\
	Because the model contains two kinds of drones working in different ways, the model is widely applicable to fixed-point monitoring problems. It can make satisfactory planning for monitoring equipment determination questions.
\end{itemize}

\subsection{Weakness}
\begin{itemize}
	\item \textbf{Two Dimensional Approximation}\\
	The model in this article is built on a two-dimensional plane, so the application scenarios are limited. In actual flight, UAVs mostly needs to use a three-dimensional space model.
	\item \textbf{Unable to Cover Extreme Cases}\\
	We make the plan based on the worst year. But in reality, extreme wildfire disasters that are more severe than 2019 may suddenly occur. In this case, the number of drones purchased by "Rapid Bushfire Response" may not be enough and needs to be supplemented. Therefore, the model lacks a certain degree of flexibility for emergencies.
	\item \textbf{Lack of Experimental Test}\\
	Some assumptions in the model are set subjectively. So it will cause the result to lose a certain degree of accuracy, which may be quite different from the actual situation. In the future, we hope to be able to carry out physical experiments to optimize the model in this article.
	\item \textbf{It Is A Bit Difficult to Solve}\\
	The model involves many variables, and the variables are coupled with each other. The space complexity of model solution is related to the number of fire points. So modeling, especially the model solution process, has a large workload. This makes it difficult to expand the model, so this model still has room for further optimization.
\end{itemize}

\newpage

\section{Budget Request}

\noindent Victoria State Government:

With the constant changes in the global climate, extreme wildfire disasters occur frequently in recent years in Victoria. In order to further strengthen Victoria's wildfire control and build a complete wildfire disaster emergency management system, we, Victoria's Country Fire Authority (CFA) plan to establish a new department responsible for work related to wildfire emergency in Victoria, called "Rapid Bushfire Response".

The "Rapid Bushfire Response" department needs to be equipped with a sufficient number of SSA drones and Radio Repeater drones. The prototypes of both drones are Akme Corporation's prototype WileE–15.2X hybrid drone. SSA drones are equipped with video or telemetry, which is used to timely monitor the wearable devices on frontline personnel of the fire and transmit the data back to the Emergency Operations Center (EOC). Radio Repeater drones are equipped with radios to extend the signal range of handheld two-way radios for frontline personnel. These two types of drones help EOC obtain timely front-line information about the fire, assist EOC's command, and ensure the safety of front-line personnel.

The worst wildfires from 2000 to 2020 occurred in 2019. Adhering to the concept of "saturation rescue", we choose the situation in 2019 to be the standard. We believe that the number of SSA drones and Radio Repeater drones needs to be sufficient for this year's rescue.

In order to maximize the utilization of the budget, we try to find the minimum budget required by the "Rapid Bushfire Response" department. The cost of an SSA drones and a Radio Repeater drones are both \$10,000, so the cost of purchasing a drone is related to the number of drones required. Because the cost of SSA drones and the cost of Radio Repeater drones are independent of each other, we consider the budget of the two separately.

For SSA drones, we consider the frequency and size of fires in each small block. The number of fire spots, the flying capability of the drone, the flight trajectory of the drone, the update time of the fire spot information, information delay of EOC and other influences are used to obtain the minimum required SSA drones for each block, and then add the number of SSA drones required for all small blocks to obtain a total number of SSA drones.

For Radio Repeater drones, we determine the signal range of the handheld two-way radio according to the rugged degree of each small area. In addition, the frequency and size of the fire point, the hovering time of the drone, the distribution of the fire point and other factors determines the total number of Radio Repeater drone.

In order to give full play to the emergency role of the "Rapid Bushfire Response" department as soon as possible and ensure the normal operation of emergency work, we specially apply for you to allocate the budget for the purchase of SSA drones and Radio Repeater drones:

\begin{table}[H]
	\begin{tabular}{|p{1.2cm}|p{2.2cm}|p{1.1cm}|p{1cm}|c|c|p{1.8cm}|c|p{1.8cm}|}
	\hline
	Object     & Prototype                & from             & unit price & unit & count & total amount & unit & note                          \\ \hline
	SSA drones & WileE–15.2X Hybrid Drone & Akme & 10000      & AUD  &  7548     &  7,548,000            & AUD  & video/ telemetry \\ \hline
	radios     & WileE–15.2X Hybrid Drone & Akme & 10000      & AUD  &  2706     &   2,706,000           & AUD  & radio           \\ \hline
	\multicolumn{2}{|c|}{sum}             &                  & 10000      & AUD  &  10254     &   10,254,000           & AUD  &                               \\ \hline
	\end{tabular}
\end{table}

The total is 10,254,000 Australian dollars. It is recommended to arrange it from the special budget of Victoria’s emergency management. Approval is kindly requested.

Applicant: Victoria’s Country Fire Authority

Time: 2021.2.8

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